We have constructed a sequence of solutions of the Helmholtz equation forming an orthogonal sequence on a given surface. Coefficients of these functions depend on an explicit algebraic formulae from the coefficient of the surface. Moreover, for exterior Helmholtz equation we have constructed an explicit normal derivative of the Dirichlet Green function. In the same way the Dirichlet-to-Neumann ...

The paper presents an iterative algorithm for studying a nonlinear shallow-water wave equation. The equation is written as an evolution equation, involving only first-order spatial derivatives, coupled with the Helmholtz equation. We propose a two-step iterative method that first solves the evolution equation by the implicit midpoint rule and then solves the Helmholtz equation using a three-poi...

A general symmetric Trefftz Discontinuous Galerkin method is built for solving the Helmholtz equation with piecewise constant coefficients. The construction of the corresponding local solutions to the Helmholtz equation is based on a boundary element method. A series of numerical experiments displays an excellent stability of the method relatively to the penalty parameters, and more importantly...

CLASSICAL SCATTERING OF ELECTROMAGNETIC RADIATION Light is an electromagnetic disturbance which is propagated by vector wave equations which are readily derived from Maxwell's equations. The Helmholtz wave equation results from Maxwell's equations. The Helmholtz equation is linear; thus, superposition of solutions is allowed. Huygens' principle is that a point source of light will give rise to ...

We present an iterative solution method for the discrete high wavenumber Helmholtz equation. The basic idea of the solution method, already presented in [18], is to develop a preconditioner which is based on a Helmholtz operator with a complex-valued shift, for a Krylov subspace iterative method. The preconditioner, which can be seen as a strongly damped wave equation in Fourier space, can be a...

A continuation method is presented for solving the inverse medium scattering problem of the Helmholtz equation in R. The algorithm requires only single-frequency scattering data. Using an initial guess from the Born approximation, each update is obtained via recursive linearization on the spatial frequency of a one-parameter family of plane waves by solving one forward and one adjoint problem o...

Since supp f ⊂ Ω, uθ satisfies the plain Helmholtz equation with constant wavenumber outside of Ω, the radiation condition and uθ|∂Ω = gθ. So by solving the exterior problem of the Helmholtz equation (analytically if Ω is a circle), uθ can be computed everywhere outside of Ω without prior knowledge of f . In particular, we can compute h = ∂uθ ∂ν on ∂Ω. Thus, our original problem can be rewritte...

In 1983, a preconditioner was proposed [J. Comput. Phys. 49 (1983) 443] based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is especially effective for low wavenumber cases where the linear system is slightly indefinite. Laird [Preconditioned iterative solution of the 2D Helmholtz equation, First Year’s Report, St. Hugh’s College, ...

This chapter is devoted to the investigation of the Dirichlet problem for the Helmholtz equation in a bounded domain. In Section 1, we briefly explain why and how boundary valued problems for the Helmholtz equation are solved. Using the properties ofDα, Tα, Fα as well as the projections defined by the singular Cauchy operator the Dirichlet problem for classical α−hyperholomophic functions are s...